Splitting theorems for hypersurfaces in Lorentzian manifolds

TL;DR

This paper establishes splitting theorems for certain globally hyperbolic Lorentzian manifolds with curvature bounds, using geometric maximum principles and volume maximality conditions.

Contribution

It introduces new splitting results for spacetimes with hypersurfaces under curvature and maximality assumptions, extending previous work with more elementary proofs.

Findings

Splitting theorem under existence of a maximal length ray.

Splitting theorem with volume maximality condition.

Extension of geometric maximum principle techniques.

Abstract

This paper looks at the splitting problem for globally hyperbolic spacetimes with timelike Ricci curvature bounded below containing a (spacelike, acausal, future causally complete) hypersurface with mean curvature bounded from above. For such spacetimes we show a splitting theorem under the assumption of either the existence of a ray of maximal length or a maximality condition on the volume of Lorentzian distance balls over the hypersurface. The proof of the first case follows work by Andersson, Galloway and Howard and uses their geometric maximum principle for level sets of the (Lorentzian) Busemann function. For the second case we give a more elementary proof.